FROM THREE-VALUED NESTED SETS TO INTERVAL-VALUED (OR INTUITIONISTIC) FUZZY SETS

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 7, Number 1, Pages c 2011 Institute for Scientific Computing and Information FROM THREE-VALUED NESTED SETS TO INTERVAL-VALUED (OR INTUITIONISTIC) FUZZY SETS CHENG ZHANG, QING SU, ZHIWU ZHAO, AND PANZI XIAO Abstract. In this paper,we first set up the equivalence class of three-valued nested sets based on three-valued fuzzy sets, and the operations of equivalence classes are derived from the ones of three-valued fuzzy sets. We then prove that the interval-valued (or intuitionistic) fuzzy set can be regarded as the equivalence class of three-valued nested set. Finally, we apply the above results to the research of the interval-valued convex fuzzy set. In addition, the above outcomes also show the correctness of the theory of interval-valued (/or intuitionistic) fuzzy set established by the paper [1]. Key Words. three-valued fuzzy set, three-valued nested set, intuitionistic fuzzy set, interval-valued fuzzy set. 1. Introduction In the 70s last century, the notion of a interval-valued fuzzy set has been introduced by different authors(see[2], [3] and[4])in order to formalize the vagueness. In recent years, the researches of interval-valued fuzzy set are growing, that is because in the course of applying to practice, the membership function of a fuzzy set is difficult to be determined, but the membership degree of an interval-value is relatively easy to be determined. Besides, the results of fuzzy inference with interval-valued fuzzy sets can be better to reflect the ambiguity of everyday reasoning. In 1985, the interval representation of language value was discussed by Schwarz in [5]. In 1986, interval-valued fuzzy sets which based on the norm were studied by Turksen in [6]. In 1987, a method about interval-valued fuzzy inference was given by Gorzalczany in [7]. In the paper [8]-[11], the basic research of interval-valued fuzzy sets were studied. In 1986, the notion of intuitionistic fuzzy sets were introduced by Atanassov in [12], from then on, the intuitionistic fuzzy groups, intuitionistic fuzzy subspaces and intuitionistic fuzzy topology appeared in [13]-[16]. Although the results on the interval-valued (/or intuitionistic) fuzzy set were richness, the nested sets theory on the interval-valued (/or intuitionistic) fuzzy set is not established similarly to the fuzzy sets, the reason lied in the interval-valued fuzzy sets cut sets which defined in the article [8]-[10]was a classical set. The similar case appears in the theory of the intuitionistic fuzzy sets, which leads to nested sets theory for the interval-valued (/or intuitionistic) fuzzy sets that have not been established so far. We know that a fuzzy set can be seen as a equivalence class of a nested set [17]. In the fuzzy sets theory, we can define a fuzzy set according to a equivalence class of classical nested set, and the operations of the fuzzy sets according to the Received by the editors June 10, 2010 and, in revised form, July 22, Mathematics Subject Classification. 93A30. This research was supported in part by National Natural Science Foundation of China (Major Research Plan, No ). 11

2 12 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO operations and, or, not of classical sets. What s more, we can also obtain the definition of the membership function of the fuzzy sets via characteristic function. In this manner, we accomplish the extension from boolean algebra to soft algebra. Meanwhile, a problem occurs: whether a interval-valued fuzzy set can be seen as a equivalence class of some nested sets. Therefore, we begin our work for the interest of this problem. The paper is organized as follows. In section 2, we present three-valued nested sets and their operations. In section 3, we establish equivalence classes of three-valued nested sets and present their operations. In section 4, we construct a isomorphism between the interval-valued (/or intuitionistic) fuzzy set system and the three-valued nested set system. In section 5, we put the above theory into interval-valued (/or intuitionistic) convex fuzzy sets research. 2. Three-valued nested sets and its operation Let X be a set. The mapping A : X {0, 1 2,1} is called a three-valued fuzzy set on X. A class of all three-valued fuzzy sets over X is denoted by 3 X. For any A,B,A t 3 X,x X, the operations are defined as follows: (A B)(x) = A(x) B(x); (A B(x) = A(x) B(x); (ca)(x) = 1 A(x); ( A t )(x) = A t (x); ( A t )(x) = A t (x); X(x) 1, x X; (x) 0, x X. Then (3 X,,,C,X, )is an F lattice. Definition 2.1. [1] A mapping H : [0,1] 3 X,λ H(λ) is called a three-valued nested set of X if H satisfies that H(λ 1 ) H(λ 2 ), whenever λ 1,λ 2 [0,1] and λ 1 < λ 2. We use IU denote the class of all the three-valued nested sets over X. There exist maximum element X and minimum element,x(λ) X, (λ), which satisfy X H = X,X H = H, H = H, H =. Definition 2.2. [1] For H i IU(X)(i = 1,2),H t IU(X)(t T), the operations are defined as follows: H 1 H 2 : (H 1 H 2 )(λ) = H 1 (λ) H 2 (λ); H 1 H 2 : (H 1 H 2 )(λ) = H 1 (λ) H 2 (λ); CH : (CH)(λ) = CH(1 λ);

3 FROM THREE-VALUED NESTED SETS TO IVFS 13 H t : ( H t )(λ) = H t (λ); H t : ( H t )(λ) = H t (λ). It is not difficult to verify that H 1 H 2,H 1 H 2,CH, three-valued nested sets. H t and Proposition 2.1. (IU,,,C,X, ) has the properties as follows: (1) Idempotent law: H H = H,H H = H; (2) Commutative law: H 1 H 2 = H 2 H 1,H 1 H 2 = H 2 H 1 ; (3) Associative law: H 1 (H 2 H 3 ) = (H 1 H 2 ) H 3,H 1 (H 2 H 3 ) = (H 1 H 2 ) H 3 ; (4) Absorption law: H 1 (H 1 H 2 ) = H 1 (H 1 H 2 ) = H 1 ; (5) Distributive law: H 1 (H 2 H 3 ) = (H 1 H 2 ) (H 1 H 3 ), H 1 (H 2 H 3 ) = (H 1 H 2 ) (H 1 H 3 ); (6) Identity law: H X = H,H = H,H X = X,H = ; (7) Involution law: C(CH) = H; (8) De Morgan law: C( H t ) = CH t,c( H t ) = CH t ; (9) Infinitedistribute law: H ( H t ) = (H H t ),H ( H t ) = H t are also (H H t ). Proof. We only show that (8) and (9) are correct, and the others are similar. (C( H t ))(λ) = C(( H t )(1 λ)) = C( H t (1 λ)) Thus, C( H t ) = CH t. = C(H t (1 λ)) = (CH t )(λ). (H ( H t ))(λ) = H(λ) ( H t )(λ) = H(λ) ( H t (λ)) Therefore, H ( H t ) = (H H t ). = (H(λ) H t (λ)) = (H H t )(λ). The others are parallelism, and proofs are omitted. By Proposition 2.1, we have known that three-valued nested sets have similar properties to the fuzzy sets. 3. Equivalence class of three-valued nested sets Definition 3.1. A relation in IU(X) is defined as follows: for any λ (0,1]. H 1 H 2 H 1 (α) = H 2 (α)

4 14 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO Then the above is an equivalence relation, that is, it satisfies (a) reflexivity: H H; (b) symmetry: H 1 H 2 H 2 H 1 ; (c) transitivity: H 1 H 2,H 2 H 3 H 1 H 3. In IU(X), we define for all λ [0,1]. Let and H 1 H 2 H 1 (λ) H 2 (λ) [H] = {H H U(X),H H} IVF (X) IU(X)/ = {[H] H IU(X)} Lemma 3.1. Let H IU(X), we define F H : [0,1] 3 X,λ F H (λ) H(α),(λ (0,1]),F H (0) = X; F H : [0,1] 3 X,λ F H (λ) H(α),(λ [0,1)),F H (1) =. Then (1) F H,F H IU(X); (2) λ 1 < λ 2 F H (λ 1 ) F H (λ 2 ); (3) F H H F H ; (4) F H (λ) = F H (α),(λ (0,1]),F H (λ) = (5) F H (λ) = F H (α),(λ (0,1]),F H (λ) = F H (α),(λ [0,1)); F H (α),(λ [0,1)). Proof. (1) If λ 1 < λ 2, then F H (λ 1 ) = H(α) H(α) = F H (λ 2 ). 1 2 (2) If λ 1 < λ 2, then there exists α 0 such that λ 1 < α 0 < λ 2, thus F H (λ 1 ) = H(α) H(α 0 ) H(α) = F H (λ 2 ). 1 2 (3) By H(α) H(λ)( α > λ), we have F H (λ) = By H(α) H(λ)( α < λ), we have F H (λ) = It follows that therefore (4) Since then F H (λ) = F H (α) = F H (λ) H(λ) F H (λ), β<α F H (α),(λ (0,1]). In the following, we will show that If λ 1, then F H (α) F H (λ) F H H F H. H(β) = β<α, F H (λ) = F H (α),(λ [0,1)). hand, F H (α) H(α) for all α [0,1], thus follows that F H (λ) = F H (α). (5) we first prove that F H (λ) = H(α) H(λ). H(α) H(λ). H(β) = β<λ H(β) = F H (λ), F H (λ) F H (λ) for all α > λ. On the other F H (α) H(α) = F H (λ), it F H (α),(λ (0,1]).

5 FROM THREE-VALUED NESTED SETS TO IVFS 15 If λ 0, then by F H (α) F H (λ) for any α < λ, we have On the other hand, F H (α) H(α) for any α [0,1], it follows that H(α) = F H (λ). Therefore F H (λ) = In the next place, we will prove that F H (λ) = In fact, F H (α) = That is, H(β) = β>α β>α,α>β Lemma 3.2. Let H 1,H 2 IU(X), then H 1 H 2 Proof. It only needs to prove that F H (α)(λ (0,1]). F H (α) F H (λ). F H (α)(λ [0,1)). F H (α) H(β) = H(β) = F H (λ)(λ [0,1)). β>λ F H (λ) = F H (α). H 1 (α) = H 2 (α)(λ [0,1)) F H1 (λ) = F H2 (λ) λ [0,1),F H1 (λ) = F H2 (λ) for all λ (0,1]. As a matter of fact, F H1 (λ) = F H2 (λ) F H1 (λ) = (0,1]; F H1 (λ) = F H2 (λ) F H1 (λ) = F H1 (α) = F H1 (α) = F H2 (α) = F H2 (λ) for all λ F H2 (α) = F H2 (λ) for all λ [0,1). Theorem 3.1. Let H,H,H t,h t IU(X)(t T) and H t H t (t T),H H, then H t H t H t,ch CH. H t, Proof. (1) According to H t H t(t T) H t (α) = H t (α)( λ (0,1]), it is obvious that in order to have that H t we only need to prove that H t ( = )(α) ( H t)(α). In fact, ( H t )(α) = H t (α) = H t (α) = H t (α) = H t (α) = ( H t )(α)(λ (0,1]). (2) By Lemma 3.2, we only need to prove that ( H t )(α) = ( H t)(α) Asamatteroffact, ( H t )(α) = H t (α)= H t (α) = H t (α) = H t (α) = ( H t )(α). (3) By lemma 3.2, H t H t(t T) H t (α) = H t(α)(λ [0,1)).

6 16 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO (CH)(α) = CH(1 α) = C( H(1 α)) = C( 1 α>1 λ = C(H (1 α)) = (CH )(α),(λ [0,1)). By Definition 3.1, we known that CH CH. 1 α>1 λ Definition 3.2. The operations,,c in IVF (X) are defined as follows: [H t ] = [ H t ], [H t ] = [ H t ],C[H] = [CH]. By Theorem 3.1, we have known that Definition 3.2 is well-defined. For A 3 X, let H A (λ) A,IP (X) = {[H A ] A 3 X }, H (1 α)) then IP (X) IVF (X),(IP (X),,,C) and (3 X,,,C) is isomorphism. Let IVF(X) = (IVF (X) IP (X)) 3 X, then we obtain a new algebra system (IVF(X),,,C), and it is isomorphic with system (IVF (X),,,C) by the embedding theorem of algebra [18]. In the following, wewillprovethat the algebrasystem (IVF(X),,,C) canbe isomorphic with the algebra system which formed by interval-valued fuzzy sets (/or intuitionistic) fuzzy sets. So we can regard the algebra system (IVF(X),,,C) as the model of interval-valued (/or intuitionist) fuzzy sets model. 4. Membership functions of interval-valued(/or intuitionistic)fuzzy sets Let L = {(α,β) α,β [0,1],α β}, L X = {A A : X L is a mapping}. A L X is called a interval-valued fuzzy set and denoted by A = [A,A + ], which A : X [0,1],A + : X [0,1], moreover A (X) A + (X), x X. Definition 4.1. For any A,B,A (γ) L X, let A = [A,A + ],B = [B,B + ],A (γ) = [A (γ),a+ (γ)], we define A B A (x) B (X),A + (X) B + (X), x X; A = B A (X) = B (X),A + (X) = B + (X), x X; CA = [1 A +,1 A ]; (γ) = γ ΓA [ γ ΓA (γ), A + (γ) ]; γ Γ (γ) = γ ΓA [ γ ΓA (γ), A + (γ) ]. γ Γ Theorem 4.1. If mapping f : IVF(X) L X,A = [H] [A,A + ]. where A (x) = {λ H(λ)(x) = 1},A + (x) = {λ H(λ)(x) 1 2 }. Then (1) f is a bijective. (2) A λ = {x F H(λ)(x) = 1},A + λ = {x F H(λ)(x) 1 2 }, A λ = {x F H(λ)(x) = 1},A + λ = {x F H(λ)(x) 1 2 }. (3) f keeps operations, that is f( [H t ]) = f([h t ]),f( t ]) = [H f([h t ]),f(c[h]) = Cf([H])

7 FROM THREE-VALUED NESTED SETS TO IVFS 17 Proof. (1) Firstly, we need to prove that f is well-defined. Suppose that A = [H] IVF(X). By Lemma 3.1, for H [H] we have It follows that F H (λ) = F H (λ) H (λ) F H (λ) = F H (λ)( λ [0,1]). {λ F H (λ)(x) = 1} {λ H(λ)(x) = 1} {λ F H (λ)(x) = 1}, {λ F H (λ)(x) 1 2 } {λ H(λ)(x) 1 2 } {λ F H(λ) 1 2 }. For any λ {α F H (α)(x) = 1}, we have F H (λ )(x) = 1. Since F H (λ) F H (λ ) for λ < λ, thus F H (λ)(x) F H (λ )(x), and then F H (λ)(x) = 1, therefore, {λ F H (λ)(x) = 1} {λ λ [0,1],λ < λ } = λ, consequently {λ F H (λ)(x) = 1} {α F H (α)(x) = 1}, therefore {λ F H (λ)(x) = 1} = {λ F H (λ)(x) = 1}. Analogously, λ {α F H (α)(x) 1 2 }, we have F H(λ )(x) 1 2 for any λ < λ, and then F H (λ) F H (λ ), so F H (λ)(x) F H (λ )(x) 1 2, thus {λ F H (λ)(x) 1 2 } {λ λ [0,1],λ < λ } = λ, which implies that {λ F H (λ)(x) 1 2 } {α F H (α)(x) 1 2 }. It follows that {λ F H (λ)(x) 1 2 } = {λ F H(λ)(x) 1 2 }. Therefore f is well-defined, and furthermore, A (x) = {λ F H (λ)(x) = 1} = {λ H(λ)(x) = 1} = {λ F H (λ)(x) = 1}, A + (x) = {λ F H (λ)(x) 1 2 } = {λ H(λ)(x) 1 2 } = {λ F H(λ)(x) 1 2 }. (2) If F H (λ)(x) = 1, then A (x) = {λ F H (λ)(x) = 1} λ; If F H (λ)(x) 1, then F H (λ)(x) = ( H(α))(x) = H(α)(x) 1, thus there exists α 0 < λ, such that H(α 0 )(x) 1. For all λ α 0, we have H(λ ) H(α 0 ), then H(λ )(x) 1, so {λ H(λ)(x) = 1} {λ λ α 0 }, this leads to a contradiction which implies that A (x) = {λ H(λ)(x) = 1} {λ λ α 0 } = α 0 < λ, A λ = {x F H(λ)(x) = 1}. Similarly, we have that A + λ = {x F H(λ)(x) 1 2 },A λ = {x F H (λ)(x) = 1} and A + λ = {x F H (λ)(x) 1 2 }. (3) We will prove that f is to be a bijective. On the one hand, suppose A L X, put 1, x A λ 1 H(λ)(x) = 2, x A+ λ A λ 0, x / A + λ then H(λ) is a three-valued fuzzy sets. According to A (x) = {λ H(λ)(x) = 1} = {λ x A λ },A+ (x) = {λ H(λ)(x) 1 2 } = {λ x A+ λ }, we know that f([h]) = A = [A,A + ], thus f is a surjection. On the other hand, if f([h]) = f([h ]) = A = [A,A + ], then A (x) = {λ H(λ)(x) = 1} = {λ H (λ)(x) = 1},

8 18 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO Thus for any x X, we have A + (x) = {λ H(λ)(x) 1 2 } = {λ H (λ)(x) 1 2 }. F H (λ)(x) = 1 x A λ F H (λ)(x) = 1,F H(λ)(x) = 0 x / A + λ F H (λ)(x) = 0. Therefore H(α) = F H (λ) = F H (λ) = H (α)(λ [0,1]), i.e., [H] = [H ], which implies that f is a injection. Consequently, f is a bijection. (4) we will prove them respectively. (a) Let f([h t ]) = [A t,a+ t ], f([h]) = [ A t, A + t ], then we have that F H (λ)(x) = 1 x ( A t ) λ t T,F Ht (λ)(x) = 1 ( F Ht )(λ)(x) = 1 and F H (λ)(x) = 0 x / ( A + t ) λ ( A + t )(x) < λ t 0 T,(A + t 0 )(x) < λ F Ht0 (λ)(x) = 0 F Ht (λ)(x) = 0 ( F Ht (λ))(x) = 0. Then F H (λ)(x) = ( F Ht (λ))(x), i.e., F H (λ) = ( F Ht )(λ), thus [H] = [F H ] = [ F Ht ] = [F Ht ] = [H t ], itfollowsthatf( [H t ]) = f([h]) = [ A t, A + t ] = [A t,a+ t ] = f([h t ]). (b)leth,h t IU(X)(t T)andf([H t ]) = [A t,a+ t ],f([h]) = [ A t, A + t ]. Then we have that F H (λ)(x) = 1 x ( A t ) λ t 0 T,F Ht0 (λ)(x) = 1 ( F Ht )(λ)(x) = 1 and F H (λ)(x) = 0 x / ( A + t ) λ A + t (x) λ t T,(A + t )(x) < λ t T,F Ht (λ)(x) = 0 F Ht (λ)(x) = 0 ( F Ht (λ))(x) = 0. Thus F H (λ)(x) = ( F Ht (λ))(x), i.e, F H (λ) = ( F Ht )(λ). Hence [H] = [F H ] = [ F Ht ] = [F Ht ] = [H t ]. It follows that f( [H t ]) = f([h]) = [ A t, A + t ] = [A t,a + t ] = f([h t ]). (c) Let f([h]) = [A,A + ], [H] IU(X) and f([h]) = [CA +,CA ]. Then we get F H (λ)(x) = 1 x (CA + ) λ (CA + )(x) λ 1 A + (x) λ A + (x) 1 λ x / A + 1 λ F H (1 λ)(x) = 0 CF H (1 λ)(x) = 1 (CF H )(λ)(x) = 1 and F H (λ)(x) = 0 x / (CA ) λ (CA )(x) < λ 1 A (x) < λ A (x) > 1 λ x A 1 λ F H (1 λ)(x) = 1 CF H (1 λ)(x) = 0 (CF H )(λ)(x) = 0. Thus F H (λ) = (CF H )(λ), therefore [H] = [F H ] = [CF H ] = C[F H ] = C[H]. It follows that f(c[h]) = f[h] = C(f([H])). By Theorem 4.1, it is obvious that the system (IVF(X),,,C) is isomorphic with the system (L X,,,C). Hence we need not to distinguish A = [H] from f([h]) = [A,A + ], and call [A (x),a + (x)] as the membership interval of x for A. If we consider the interval-valued fuzzy set A = [H] and the membership interval of A as the same, we can obtain the definition of Zadeh interval-valued fuzzy sets. Therefore, from the viewpoint of the mathematics, an interval-valued fuzzy set can be seen as an equivalent class, which illustrates the expansion from three-valued case to interval-valued fuzzy case. Let L = {(a,b) a,b [0,1],a+b 1)}, we define

9 FROM THREE-VALUED NESTED SETS TO IVFS 19 (a) (a 1,b 1 ) (a 2,b 2 ) a 1 a 2,b 1 b 2 ; (b) (a 1,b 1 ) (a 2,b 2 ) = (a 1 a 2,b 1 b 2 ),(a 1,b 1 ) (a 2,b 2 ) = (a 1 a 2,b 1 b 2 ); (c) (a,b) c = (b,a); (d) (a t,b t ) = ( a t, b t ); (a t,b t ) = ( a t, b t ); (e) 1 = (1,0),0 = (0,1). Suppose L X = {A A : X L is a mapping }, and A L X. A is said to be a intuitionistic fuzzy set and denoted by A = (µ A,ν A ), where µ A : X [0,1],ν A : X [0, 1] represent the degree of membership and nonmembership, respectively, satisfying µ A (x)+ν A (x) 1, x X. Let IF(X) denote the class of all the intuitionistic fuzzy sets over X, the operations in IF(X) are derived by the operations in L as follows: Let A = (µ A,ν A ),B = (µ B,ν B ),A t = (µ At,ν At ),(t T), then (1) A B µ A (x) µ B (x),ν A (x) ν B (x), x X; (2) CA = (ν A,µ A ); (3) A B = (µ A B,ν A B ),where µ A B = µ A (x) µ B (x),ν A B = µ A (x) µ B (x); (4) A B = (µ A B,ν A B ),where µ A B = µ A (x) µ B (x),ν A B = µ A (x) µ B (x); (5) Let A = A t, where µ A (x) = µ At (x),ν A (x) = ν At (x); (6) Let B = A t, where µ B (x) = µ At (x),ν B (x) = ν At (x); (7) Let X(x) (1,0), x X; (x) = (0,1), x X. (IF(X),,,C, X, ) is a fuzzy lattice. We define a mapping between IF(X) and IVF(X) by ϕ : A = (µ A,ν A ) [µ A,Cν A ], then the mapping ϕ is a isomorphism of algebraic systems (IF(X),,,C) and (IVF(X),,,C). By the Theorem 4.1, we have Theorem 4.2. If mapping h : IVF(X) L X,A = [H] (µ A,ν A ), where µ A (x) = {λ H(λ)(x) = 1},ν A (x) = {1 λ H(λ)(x) 1 2 }, then (1) h is a bijective; (2) (µ A ) λ = {x F H (λ)(x) = 1},(ν A ) λ = {x F H (1 λ)(x) = 0}, (µ A ) λ = {x F H (λ)(x) = 1},(ν A ) λ = {x F H (1 λ)(x) = 0}. (3) h keeps the operations, i.e., h( [H t ]) = h([h t ]),h( [H t ]) = h([h t ]),h(c[h]) = Ch([H]). 5. Interval-valued(/or intuitionistic) convex fuzzy sets Suppose X is a vector space, let A = [A,A + ] be a interval-valued fuzzy set of X. By Theorem 4.1, we have that the algebra system (IVF,,,C) is isomorphic with the algebra system (L X,,,C) (/or (L X,,,C)), therefore, there exists [H] IVF(X) such that A (x) = {λ H(λ)(x) = 1}, A + (x) = {λ H(λ)(x) 1 2 }. A = [A,A + ] is said to be a interval-valued convex fuzzy set over X if there exists a three-valued nested set H in the equivalence class [H] such that H(λ) is a three-valued convex fuzzy set. Theorem 5.1. The following assertions are equivalent. (1) A = [A,A + ] is an interval-valued convex fuzzy set in X; (2) A λ anda+ λ are convex sets for arbitrary λ (0,1]; (3) For arbitrary a [0,1] and x,y X, then A (ax+(1 a)y) min{a (x),a (y)},a + (ax+(1 a)y) min{a + (x),a + (y)}; (4) A λ,a+ λ are convex sets for arbitrary λ (0,1];

10 20 C. ZHANG, Q. SU, Z. ZHAO AND P. XIAO (5) For arbitrary x 1,,x p E and all a 1,,a p [0,1] which make p a i = 1, we have A ( p a i x i ) min(a (x 1 ), A (x p )),A + ( p a i x i ) min(a + (x 1 ), A + (x p )). i=1 The similar conclusions can be acquired on intuitionistic fuzzy sets by Theorem 4.2, and we omit them in this paper. 6. Conclusions Through the research of three-valued fuzzy sets and three-valued nested sets in paper[1], the definition the interval-valued(/or intuitionistic) fuzzy set is formulated. Based on the equivalence class of three-valued fuzzy nested sets and the operations of interval-valued(/or intuitionistic) fuzzy sets also are formed, which illustrated the expansion from fuzzy sets to interval-valued(/or intuitionistic) fuzzy sets and supplied the theoretical basis for paper[1] from another aspect. Furthermore, as the application of the above theory, we discussed the interval-valued(/or intuitionistic) convex fuzzy set, and the derived results show that when the intervalvalued(/or intuitionistic) fuzzy set degenerate the fuzzy set, the above theory is consistent with fuzzy set theory. However, we can not obtain the above ones using the previous concept of the interval-valued fuzzy set s cut set. It shows that the definition of the cut set defined in the paper [1] is reasonable. By analogy with the fuzzy set, we will try to explore the algebra, analysis, optimization and decision theory of the interval-valued(/or intuitionistic) fuzzy set. Acknowledgments The author thanks the anonymous authors whose work largely constitutes this sample file. This research was supported by in part by National Natural Science Foundation of China (Major Research Plan, No ). References [1] Yuan Xuehai,Li Hong xing,sun Kaibiao. The cut set of Intuitionistic fuzzy sets and intervalvalued fuzzy sets and decomposition theorem and the Representation Theorem, Science in China (E Series),2009,39(9): [2] Zadeh L A., Outline of a new approach to the analysis of complex systems and decision processes, interval-valued fuzzy sets, IEEE Trans. Systems Man Cybernet.1973, 3(1): [3] Grattan-Guiness, I., Fuzzy membership mapped onto interval and many valued quantities, Z. Math. Logik Grundlag. Math., 22(1975), [4] Jahn, K. U., Interval-wertige mengen. Math. Nachr., 68(1975), [5] Schwarz D. G., The case for an interval based representation of ligiistic truth, Fuzzy Sets and Systems, 20(1985), [6] Turksen I.B., Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, 20(1986), [7] Gorzalczany M.B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, 21(1987), [8] Meng Guangwu,. The Basic Theory of Interval-valued Fuzzy sets. Applied Mathematics(in Chinese), 6(1993), No.2: [9] Zeng Wenyi,Li Hongxing,Shi Yu. decomposition theorem of Interval-valued fuzzy sets, Journal of Beijing Normal University(in Chinese), 2003,39(2): [10] Zeng Wenyi,Li Hongxing,Shi Yu. Representation theorem of Interval-valued fuzzy sets. Journal of Beijing Normal University(in Chinese), 2003,39(4): [11] Glad Deschrijver, Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems, 160 (2009) [12] Atanassov K. Intuitionistic fuzzy Sets.Fuzzy Sets and Systems, 1986, 20: [13] Biswas R. Intuitionistic fuzzy subgroups, Math.Fortum, 1989,10: i=1 i=1

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